一般共生拓扑结构的概念首先是由Császár在他的专著^[1]中提出的.它综合了拓扑空间、一致结构和临近空间, 在拓扑学研究中占有重要地位. Katsaras在文[2]中把它推广到了模糊空间中, 并做了大量重要的工作^[2-5].

格模糊拓扑空间以及光滑拓扑空间中的诱导理论都有相应的讨论^{[6, 9-10]}.本文研究模糊拓扑生成序与其诱导的$I$-fuzzy拓扑生成序的关系.文中首先通过三种Lowen映射得到三种拓扑生成序.其次, 给出弱诱导共生拓扑序空间和满层共生序空间的概念, 并建立诱导共生拓扑序空间的基本理论.最后, 文中讨论诱导$I$-fuzzy拓扑与诱导的I-fuzzy拓扑生成序的从属关系.

称$X$上的二元映射$\tau:X\times X\rightarrow I$为一个fuzzy拓扑生成序, 若$\tau$满足以下条件:

(FTO 1) $\tau(\emptyset, \emptyset)=\tau(X, X)=1$.

(FTO 2) 若$\tau(U, V)>0$, 则$U\subset V$.

(FTO 3) 若$U\subset U_1$, $V_1\subset V$, 则$\tau(U_1, V_1)\leq\tau(U, V)$.

(FTO 4) $\tau(U\cup U_1, V)\geq\tau(U, V)\wedge\tau(U_1, V)$.

(FTO 5) $\tau(U, V\cap V_1)\geq\tau(U, V)\wedge\tau(U, V_1)$.

称Fuzzy拓扑生成序$\tau$为双完全的, 若$\tau$还满足:

(FTO 6) $\tau(\cup_{j\in J}U_j, V)\geq\wedge_{j\in J}\tau(U_j, V)$, $\forall\{U_j, V\subset X: j\in J\}$.

(FTO 7) $\tau(U, \cap_{j\in J}V_j)\geq\wedge_{j\in J}\tau(U, V_j)$, $\forall\{U, V_j\subset X: j\in J\}$.

若$\tau$是fuzzy(双完全)拓扑生成序, 那么称$(X, \tau)$为fuzzy(双完全)拓扑生成序空间.若$\tau_1, \tau_2$都是fuzzy(双完全)拓扑生成序, 那么$\tau_1\vee\tau_2$和$\tau_1\wedge\tau_2$也是.

设$(X, \tau_1), (Y, \tau_2)$是fuzzy拓扑生成序空间, $U, V\subset Y$.称$f:(X, \tau_1)\rightarrow(Y, \tau_2)$为

(1) 连续, 若$\tau_1(f^{-1}(U), f^{-1}(V))\geq\tau_2(U, V)$.

(2) 弱连续, 若$\tau_2(U, V)>0$, 则$\tau_1(f^{-1}(U), f^{-1}(V))>0$.\\称$I^X$上的二元映射$\eta:I^X\times I^X\rightarrow I$为$I$-fuzzy拓扑生成序, 若$\eta$满足:

(I-FTO 1) $\eta(0_X, 0_X)=\eta(1_X, 1_X)=1$.

(I-FTO 2) $\eta(A, B)>0$, 则$A\leq B$.

(I-FTO 3) 若$A\leq A_1$, $B_1\leq B$, 则$\eta(A_1, B_1)\leq\eta(A, B)$.

(I-FTO 4) $\eta(A\vee A_1, B)\geq\eta(A, B)\wedge\eta(A_1, B)$.

(I-FTO 5) $\eta(A, B\wedge B_1)\geq\eta(A, B_1)\wedge\eta(A, B_1)$.

称$I$-fuzzy拓扑生成序$\eta$是双完全的, 若$\eta$还满足:

(I-FTO 6) $\eta(\vee_{j\in J}A_j, B)\geq\wedge_{j\in J}\eta(A_j, B)$, $\forall\{A_j, B\in I^X: j\in J\}$.

(I-FTO 7) $\eta(A, \wedge_{j\in J}B_j)\geq\wedge_{j\in J}\eta(A, B_j)$, $\forall\{A, B_j\in I^X: j\in J\}$.

若$\eta$是fuzzy(双完全)拓扑生成序, 那么称$(I^X, \eta)$为$I$-fuzzy(双完全)拓扑生成序空间.若$\eta_1, \eta_2$都是fuzzy(双完全)拓扑生成序, 那么$\eta_1\vee\eta_2$和$\eta_1\wedge\eta_2$也是^[4].

称映射$f^{\rightarrow}:I^X\rightarrow Y$为模糊映射, 若存在$f:X\rightarrow Y$, 使$\forall A\in I^X, y\in Y$,

$ f^{\rightarrow}(A)(y)=\vee\{A(x):x\in X, f(x)=y\}. $

若$f^{\rightarrow}$是双射.记其逆映射$f^{\leftarrow}:I^Y\rightarrow I^X$为: $\forall B\in I^Y, x\in X$, $f^{\leftarrow}(B)(x)=B(f(x))$.若$A\in I^X$, 记$\sigma_r(A)=\{x:A(x)\geq r\}$, 则以下结论显然成立:

(1) $U\subset X$, $V\subset Y$, $f^{\rightarrow}(1_U)=1_{f(U)}$, $f^{\leftarrow}(1_V)=1_{f^{-1}(V)}$.

(2) $\sigma_r(f^{\rightarrow}(A))=f(\sigma_r(A))$, $\sigma_r(f^{\leftarrow}(B))=f^{-1}(\sigma_r(B))$.

(3) $\sigma_r(\vee_{j\in J}A_j)=\cup_{j\in J}\sigma_r(A_j)$, $\sigma_r(\wedge_{j\in J}A_j)=\cap_{j\in J}\sigma_r(A_j)$.

(4) 若$\underline{\lambda}\in I^X, \underline{\mu}\in I^Y$是常值模糊集, 则$f^{\rightarrow}(\underline{\lambda})\in I^Y$和$f^{\leftarrow}(\underline{\mu})\in I^X$都是常值的.

若$(I^X, \eta_1), (I^Y, \eta_2)$是$I$-fuzzy拓扑生成序空间, $A, B\in I^Y$.称模糊映射$f^{\rightarrow}:I^X\rightarrow I^Y$为

(1) 连续, 若$\eta_1(f^{\leftarrow}(A), f^{\leftarrow}(B))\geq\eta_2(A, B)$.

(2) 弱连续, 若$\eta_2(A, B)>0$, 则$\eta_1(f^{\leftarrow}(A), f^{\leftarrow}(B))>0$.

定理2.1  设$\tau$是$X$上的fuzzy拓扑生成序, $A, B\in I^X$.令

$ \omega(\tau)(A, B)= \wedge_{r\in I}\tau(\sigma_r(A), \sigma_r(B)). $

则$\omega(\tau)$是$I^X$上的$I$-fuzzy拓扑成序.称$\omega(\tau)$是由$\tau$生成的$I$-fuzzy拓扑生成序.

证 (I-FTO 1)  $\omega(\tau)(0_X, 0_X) =\wedge_{r\in I}\tau(\sigma_r(0_X), \sigma_r(0_X))= \wedge_{r\in I}\tau(\emptyset, \emptyset)=1$,

$ \omega(\tau)(1_X, 1_X)=\wedge_{r\in I}\tau(\sigma_r(1_X), \sigma_r(1_X))= \wedge_{r\in I}\tau(X, X)=1. $

(I-FTO 2) 若$\omega(\tau)(A, B)>0$, 则$\forall r\in I$, $\tau(\sigma_r(A), \sigma_r(B))>0$, 进而$\sigma_r(A)\subset\sigma_r(B)$.因此

$ A=\vee_{r\in I}(r\wedge1_{\sigma_r(A)})\leq \vee_{r\in I}(r\wedge1_{\sigma_r(B)})=B. $

(I-FTO 3)  若$A\leq A_1$, $B_1\leq B$.则$\forall r\in I$, $\tau(\sigma_r(A_1), \sigma_r(B_1))\leq \tau(\sigma_r(A), \sigma_r(B))$.从而

$ \wedge_{r\in I}\tau(\sigma_r(A_1), \sigma_r(B_1)) \leq\wedge_{r\in I}\tau(\sigma_r(A), \sigma_r(B))=\omega(A, B). $

(I-FTO 4)

$ \begin{array}{l} \;\;\;\;\omega(A\vee A_1,B)= \wedge_{r\in I}\tau(\sigma_r(A\vee A_1), \sigma_r(B))\\ = \wedge_{r\in I}\tau(\sigma_r(A)\cup\sigma_r(A_1),\sigma_r(B))\geq \wedge_{r\in I}\tau(\sigma_r(A),\sigma_r(B))\wedge \wedge_{r\in I}\tau(\sigma_r(A_1),\sigma_r(B))\\ = \omega(\tau)(A,B)\wedge\omega(\tau)(A_1,B). \end{array} $

(I-FTO 5) 类似(I-FTO 4) 可证.因此$\omega(\tau)$是$I$-fuzzy拓扑生成序.若$\tau$是双完全的, 则

(I-FTO 6) 若$\{U_j\subset X:j\in J\}$, 有

$ \begin{array}{l}\;\;\;\;\;\omega(\tau)(\vee_{j\in J}A_j, B)= \wedge_{r\in I}\tau(\sigma_r(\vee_{j\in J}A_j), \sigma_r(B))\\ \geq \wedge_{r\in I}\wedge_{j\in J} \tau(\cup_{j\in J}\sigma_r(A_j), \sigma_r(B))\\ = \wedge_{j\in J}\wedge_{r\in I}\tau(\sigma_r(A_j), \sigma_r(B))= \wedge_{j\in J}\omega(\tau)(A_j, B).\end{array} $

(I-FTO 7) 类似(I-FTO 6) 可证.

说明1  称$I$-fuzzy拓扑生成序$\eta$是可生成的, 若存在$X$上的fuzzy拓扑生成序$\tau$使$\eta=\omega(\tau)$.

定理2.2(1)  若$\tau_1\leq\tau_2$, 则$\omega(\tau_1)\leq\omega(\tau_2)$.

(2) $\forall U, V\subset X$, $\omega(\tau)(1_U, 1_V)=\tau(U, V)$.

(3) 任意$\lambda$$\in I^X$, 有$\omega(\tau)($$\lambda$, $\lambda$$)=1$.

定理2.3   $f^{\rightarrow}:(I^X, \omega(\tau_1)) \rightarrow(I^Y, \omega(\tau_2))$ (弱)连续当且仅当$f:(X, \tau_1) \rightarrow(Y, \tau_2)$(弱)连续.

定理2.4  设$(I^X, \eta)$是$I$-fuzzy(双完全)拓扑生成序, $\forall U,V \subset X$, 记$[\eta](U, V)=\eta(1_U, 1_V)$, 则$[\eta]$是$X$上的fuzzy(双完全)拓扑生成序.

证 (FTO 1) $[\eta](\emptyset, \emptyset)=\eta(0_X, 0_X)=1$. $[\eta](X, X)=\eta(1_X, 1_X)=1$.

(FTO 2) $\forall U, V\subset X$, 若$[\eta](U, V)>0$, 则$\eta(1_U, 1_V)>0$.则$1_U\leq1_V$.于是$U\subset V$.

(FTO 3) 若$U\subset U_1$, $V_1\subset V$, 有$[\eta](U_1, V_1)=\eta(1_{U_1}, 1_{V_1})\leq\eta(1_U, 1_V)=[\eta](U, V)$.

(FTO 4) $[\eta](U\cup U_1, V)=\eta(1_U\vee1_{U_1}, 1_V) \geq\eta(1_U, 1_V)\wedge\eta(1_{U_1}, 1_V) =[\eta](U, V)\wedge[\eta](U_1, V)$.

(FTO 5) $[\eta](U, V\cap V_1)=\eta(1_U, 1_V\wedge1_{V_1}) \geq\eta(1_U, 1_V)\wedge\eta(1_U, 1_{V_1}) =[\eta](U, V)\wedge[\eta](U, V_1)$.

(FTO 6) $[\eta](\cup_{j\in J}U_j, V)=\eta(1_{\cup_{j\in J}U_j}, 1_V) =\eta(\vee_{j\in J}1_{U_j}, 1_V)\geq \wedge_{j\in J}[\eta](U_j, V)$.

(FTO 7) $[\eta](U, \cap_{j\in J}V_j)=\eta(1_U, 1_{\cap_{j\in J}V_j}) =\eta(1_U, \wedge_{j\in J}1_{V_j})\geq \wedge_{j\in J}[\eta](U, V_j)$.

定理2.5  若$f^{\rightarrow}:(I^X, \eta_1)\rightarrow(I^Y, \eta_2)$(弱)连续, 则$f:(X, [\eta_1])\rightarrow(Y, [\eta_2])$也(弱)连续.

定理2.6  设$(I^X, \eta)$是$I$-fuzzy(双完全)拓扑生成序, $\forall U, V\subset X, r\in I$, 记

$ \iota_r(\eta)(U, V)=\vee\{\eta(A, B):\sigma_r(A, B)=(U, V)\}, $

则$\iota_r(\eta)$是fuzzy(双完全)拓扑生成序.

证 (FTO 1) $\iota_r(\eta)(\emptyset, \emptyset)\geq\eta(0_X, 0_X)=1$, $\iota_r(\eta)(X, X)\geq\eta(1_X, 1_X)=1$.

(FTO 2) 若$0<\iota_r(\eta)(U, V)=\vee\{\eta(A, B):\sigma_r(A, B)=(U, V)\}$.存在$A, B\in I^X$, 使$\sigma_r(A, B)=(U, V)$, 并且$\eta(A, B)>0$.于是$A\leq B$.从而$U=\sigma_r(A)\subset\sigma_r(B)=V$.

(FTO 3) 若$U\subset U_1$, $V_1\subset V$, 有

$ \begin{array}{l} \iota_r(\eta)(U_1, V_1)=\vee\{\eta(A_1, B_1):\sigma_r(A_1, B_1) = (U_1, V_1)\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq \vee\{\eta(A, B):\sigma_r(A, B)=(U, V)\}=\iota_r(\eta)(U, V).\end{array} $

(FTO 4) $\forall U, U_1, V\subset X, \varepsilon>0$, 存在$A, A_1, B, D\in I^X$, 使

$ \sigma_r(A, A_1)=(U, U_1), \sigma(B, D)=(V, V), $

且$\iota_r(\eta)(U, V)-\varepsilon<\eta(A, B)$, $\iota_r(\eta)(U_1, V)-\varepsilon<\eta(A_1, D)$.因

$ \sigma_r(A\vee A_1)=U\cup U_1, \sigma_r(B\vee D)=V, $

则

$ \begin{array}{l} \eta(A\vee A_1, B\vee D)\geq \eta(A, B)\wedge\eta(A_1, D) >(\iota_r(\eta)(U, V)-\varepsilon)\wedge(\iota_r(\eta)(U_1, V)-\varepsilon)\\ =\iota_r(\eta)(U, V)\wedge\iota_r(\eta)(U_1, V)-\varepsilon.\end{array} $

由$\varepsilon$的任意性, 得

$ \eta(A\vee A_1, B)\geq\eta(A, B)\wedge\eta(A_1, B) >\iota_r(\eta)(U, V)\wedge\iota_r(\eta)(U_1, V). $

从而

$ \iota_r(\eta)(U\vee U_1, V)\geq\eta(A\vee A_1, B)\geq\iota_r(\eta)(U, V)\wedge\iota_r(\eta)(U_1, V). $

(FTO 5) 类似(FTO 4) 可证.另外, 若$\eta$双完全, $\forall U_j\subset X, j\in J$, 有

(FTO 6) $U_j, V\subset X, j\in J, \varepsilon>0$, 存在$A_j, B\in I^X$, 使$\iota_r(\eta)(U_j, V)-\varepsilon\leq\eta(A_j, B)$.于是

$ \iota_r(\eta)(\cup_{j\in J}U_j, V)\geq\wedge_{j\in J}\eta(A_j, B) \geq\wedge_{j\in J}(\iota_r(\eta)(U_j, V)-\varepsilon)= \wedge_{j\in J}\iota_r(\eta)(U_j, V)-\varepsilon. $

由$\varepsilon$的任意性可知$\iota_r(\eta)(\cup_{j\in J}U_j, V)\geq \wedge_{j\in J}\iota_r(\eta)(U_j, V)$.类似可证(FTO 7).

定理2.7  设$(I^X, \eta)$是$I$-fuzzy(双完全)生成序空间.则$\iota(\eta)=\vee_{r\in I}\iota_r(\eta)$是fuzzy(双完全)生成序.

定理2.8  设$f^{\rightarrow}:(I^X, \eta_1)\rightarrow(I^Y, \eta_2)$ (弱)连续, 则$f:(X, \iota(\eta_1))\rightarrow(Y, \iota(\eta_2))$也(弱)连续.

定理2.9  设$\tau$是$X$上的fuzzy拓扑生成序, $\eta$是$I^X$上的$I$-fuzzy扑生成序.则以下结论成立:

(1) $[\omega(\tau)]=\tau$, $[\eta]\leq\iota(\eta)$.

(2) $\omega([\eta])(1_U, 1_V)=\eta(1_U, 1_V)$.

(3) $\wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}) =\omega([\eta])(A, B)$.

(4) $\omega(\iota(\eta))\geq\eta$, $\iota(\omega(\tau))=\tau$.

(5) $\omega([\eta_1\wedge\eta_2])=\omega([\eta_1])\wedge\omega([\eta_2])$, $\omega([\eta_1\vee\eta_2])=\omega([\eta_1])\vee\omega([\eta_2])$.

(6) $\omega(\iota(\eta_1\wedge\eta_2))=\omega(\iota(\eta_1))\wedge\omega(\iota(\eta_2))$, $\omega(\iota(\eta_1\vee\eta_2))=\omega(\iota(\eta_1))\vee\omega(\iota(\eta_2))$.

(7) $\omega([\omega([\eta])])=\omega([\eta])$, $\omega(\iota(\omega(\iota(\eta))))=\omega(\iota(\eta))$.

(8) $\omega([\omega(\iota(\eta))])=\omega(\iota(\eta))$, $\omega(\iota(\omega([\eta])))=\omega([\eta])$.

说明2  由定理2.9(7) 和(8) 知, 对给定的$\eta$实施$\omega, \iota$和$[ ]$运算, 至多可得四个拓扑生成序, 分别是$[\eta], \iota(\eta), \omega([\eta])$和$\omega(\iota(\eta))$.

定理2.10  设$f:X\rightarrow Y$为满射, $(X, \tau)$是fuzzy (双完全)拓扑生成序空间, 则$\tau/f=\tau\circ f^{-1}$是$Y$上的fuzzy(双完全)拓扑生成序, 并且$\omega(\tau/f)=\omega(\tau)/f^{\rightarrow}$, 这里$\omega(\tau)/f^{\rightarrow}=\omega(\tau)\circ f^{\leftarrow}$.

证  由定义易证$\tau/f=\tau\circ f^{-1}$是$Y$上的fuzzy(双完全)拓扑生成序.另外, $\forall A\in I^Y$, 有

$ \omega(\tau/f)(A)=\wedge_{r\in I}\tau/f(\sigma_r(A)) =\wedge_{r\in I}\tau(\sigma_r(f^{\leftarrow}(A)))=\omega(\tau)(f^{\leftarrow}(A)) =\omega(\tau)/f^{\rightarrow}(A). $

定理2.11   设$f:X\rightarrow Y$是双射, $(Y, \tau)$ fuzzy(双完全)拓扑生成序空间, 则$f^{-1}(\tau)=\tau\circ f$是$X$上的fuzzy(双完全)拓扑生成序, 且$f^{\leftarrow}(\omega(\tau))=\omega(f^{-1}(\tau))$, 这里$f^{\leftarrow}(\omega(\tau))=\omega(\tau)\circ f^{\rightarrow}$.

证   $\forall A, B\in I^X$,

$ \begin{array}{l} \;\;\;\;\;f^{\leftarrow}(\omega(\tau))(A, B)=\omega(\tau)(f^{\rightarrow}(A), f^{\rightarrow}(B)) =\wedge_{r\in I}\tau(\sigma_r(f^{\rightarrow}(A)), \sigma_r(f^{\rightarrow}(B)))\\ = \wedge_{r\in I}\tau(f(\sigma_r(A)), f(\sigma_r(B)))\\ = \wedge_{r\in I}f^{-1}(\tau)(\sigma_r(A), \sigma_r(B)) =\omega(f^{-1}(\tau))(A, B).\end{array} $

定理2.12  设$f^{\rightarrow}:I^X\rightarrow I^Y$满射, $(I^X, \eta)$是$I$-fuzzy(双完全)拓扑生成序空间.则$\eta/f^{\rightarrow}=\eta\circ f^{\leftarrow}$是$I^Y$上$I$-fuzzy(双完全)拓扑生成序空间.

推论2.1  设$f^{\rightarrow}:I^X\rightarrow I^Y$为满射, $(I^X, \eta)$是$I$-fuzzy拓扑生成序.则

(1) $[\eta/f^{\rightarrow}]=[\eta]/f$.

(2) $\iota(\eta/f^{\rightarrow})=\iota(\eta)/f$.

(3) $\omega([\eta/f^{\rightarrow}])=\omega([\eta])/f^{\rightarrow}$.

(4) $\omega(\iota(\eta/f))=\omega(\iota(\eta))/f^{\rightarrow}$.

定理2.13  设$f^{\rightarrow}:I^X\rightarrow I^Y$为双射, $(I^Y, \eta)$为$I$-fuzzy(双完全)拓扑生成序空间.则$f^{\leftarrow}(\eta)=\eta\circ f^{\rightarrow}$是$I^X$上的$I$-fuzzy(双完全)拓扑生成序.

推论2.2  设$f^{\rightarrow}:I^X\rightarrow I^Y$为双射, $(I^Y, \eta)$为$I$-fuzzy拓扑生成序空间.则

(1) $f^{-1}([\eta])=[f^{\leftarrow}(\eta)]$.

(2) $f^{-1}(\iota(\eta))=\iota(f^{\leftarrow}(\eta))$.

(3) $\omega([f^{\leftarrow}(\eta)])=f^{\leftarrow}(\omega([\eta]))$.

(4) $\omega(\iota(f^{\leftarrow}(\eta)))=f^{\leftarrow}(\omega(\iota(\eta)))$.

定义3.1   称$I$-fuzzy拓扑生成序空间$(I^X, \eta)$是诱导的, 若$A, B\in I^X$,

$ \wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)})=\eta(A, B). $

称$(I^X, \eta)$为弱诱导的, 若$\wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)})\geq\eta(A, B)$.

定理3.1   $I$-fuzzy拓扑生成序空间$(I^X, \eta)$是诱导的当且仅当它是可生成的.

证 必要性   设$\tau=[\eta]$.对于任意$A, B\in I^X$,

$ \begin{array}{l}\;\;\;\;\; \eta(A, B)=\wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)})= \wedge_{r\in I}[\eta](\sigma_r(A), \sigma_r(B))\\ = \wedge_{r\in I}\tau(\sigma_r(A), \sigma_r(B))=\omega(\tau)(A, B).\end{array} $

充分性  设$\eta=\omega(\tau)$. $\forall A, B\in I^X$, 则

$ \begin{array}{l}\;\;\;\;\; \eta(A, B)=\omega(\tau)(A, B)=\wedge_{r\in I}\tau(\sigma_r(A), \sigma_r(B))\\ = \wedge_{r\in I}\omega(\tau)(1_{\sigma_r(A)}, 1_{\sigma_r(B)})= \wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}).\end{array} $

故$\eta$是诱导的.

定理3.2   $I$-fuzzy拓扑生成序空间$(I^X, \eta)$是弱诱导的当且仅当$\omega([\eta])\geq\eta$.

定理3.3   $I$-fuzzy拓扑生成序$(I^X, \eta)$是弱诱导的当且仅当$[\eta]=\iota(\eta)$.

证 必要性  只要证$[\eta]\geq\iota(\eta)$.事实上, $\forall U, V\subset X$, 有

$ \begin{array}{l} \iota(\eta)(U, V) = \vee_{r\in I}\iota_r(\eta)(U, V) = \vee_{r\in I}\vee\{\eta(A, B):\sigma_r(A, B)= (U, V)\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\le\vee_{r\in I}\vee \{\wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}):\sigma_r(A, B)=(U, V)\} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \vee_{r\in I}\vee \{\eta(1_U, 1_V):\sigma_r(A, B)=(U, V)\} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \eta(1_U, 1_V)=[\eta](U, V). \end{array} $

充分性   $\forall A, B\in I^X$,

$ \begin{array}{l} \wedge_{r\in I}\eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}) = \wedge_{r\in I}[\eta](\sigma_r(A), \sigma_r(B)) = \wedge_{r\in I}\iota(\eta)(\sigma_r(A), \sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}\vee_{s\in I}\iota_s(\eta)(\sigma_r(A), \sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\geq \wedge_{r\in I}\iota_r(\eta)(\sigma_r(A), \sigma_r(B)) \geq\eta(A, B). \end{array} $

定理3.4  设$f^{\rightarrow}:I^X\rightarrow I^Y$是满射, $(I^X, \eta)$为(弱)诱导的, 则$\eta/f^{\rightarrow}$是(弱)诱导的.

定理3.5  设$f^{\rightarrow}:I^X\rightarrow I^Y$是双射, $(I^Y, \eta)$为(弱)诱导的, 则$f^{\leftarrow}(\eta)$也是(弱)诱导的.

定理3.6   设$(I^X, \eta)$为$I$-fuzzy拓扑生成序空间.记$\eta_*=\eta\wedge\omega([\eta])$, 则$\eta_*$是满足$\eta_*\leq\eta$最大的弱诱导的$I$-fuzzy拓扑生成序.

推论3.1  设$(I^X, \eta)$是弱诱导的$I$-fuzzy空间.则$\eta_*=\eta$.

定理3.7  设$(I^X, \eta)$为$I$-fuzzy拓扑生成序空间.则$\eta^*=\omega(\iota(\eta))$是满足$\eta^*\geq\eta$最小的诱导的$I$-fuzzy拓扑生成序.

证  由定理2.9知, $\eta^*$是诱导的, 且$\eta\leq\eta^*$.若$\delta$是诱导的$I$-fuzzy拓扑生成序, 且$\eta\leq\delta$.则

$ \begin{array}{l} \eta^*(A, B) = \omega(\iota(\eta))(A, B) = \wedge_{r\in I}\iota(\eta)(\sigma_r(A), \sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}\vee_{s\in I}\iota_s(\eta)(\sigma_r(A), \sigma_r(B)) \\ \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; = \wedge_{r\in I}\vee_{s\in I}\vee\{\eta(G, H):\sigma_s(G, H)=\sigma_r(A, B)\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leq \wedge_{r\in I}\vee_{s\in I}\vee\{\delta(G, H):\sigma_s(G, H)=\sigma_r(A, B)\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \wedge_{r\in I}\vee_{s\in I}\vee\{\wedge_{t\in I} \delta(1_{\sigma_t(G)}, 1_{\sigma_t(H)}):\sigma_s(G, H)=\sigma_r(A, B)\}\\ \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; \leq \wedge_{r\in I}\vee_{s\in I}\vee\{ \delta(1_{\sigma_r(G)}, 1_{\sigma_r(H)}):\sigma_s(G, H)=\sigma_r(A, B)\}\\ \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; = \wedge_{r\in I}\delta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}) = \omega([\delta])(\sigma_r(A), \sigma_r(B))=\delta(A, B). \end{array} $

推论3.2   $I$-fuzzy拓扑生成序空间$(I^X, \eta)$是诱导的当且仅当$\eta^*=\eta$.

定理3.8  设$\eta_1, \eta_2$和$\eta$都是$I^X$上的$I$-fuzzy拓扑生成序空间.则

(1) $(\eta_1\wedge\eta_2)_*=\eta_{1*}\wedge\eta_{2*}$, $(\eta_1\vee\eta_2)_*\geq\eta_{1*}\vee\eta_{2*}$.

(2) $(\eta_1\wedge\eta_2)^*=\eta_1^*\wedge\eta_2^*$, $(\eta_1\vee\eta_2)^*=\eta_1^*\vee\eta_2^*$.

(3) $\omega([\eta])_*=\omega([\eta_*])=\omega([\eta])^* =\omega(\iota(\eta_*))=\omega([\eta])$.

(4) $\omega(\iota(\eta))_*=\omega(\iota(\eta))^*= \omega(\iota(\eta^*))=\omega([\eta^*])=\omega(\iota(\eta))$.

(5) $\eta^{**}=\eta^*$, $\eta_{**}=\eta_*$.

(6) $(\eta^*)_*=((\eta^*)_*)^*=(((\eta^*)_*)^*)_*= \cdot\cdot\cdot=\omega(\iota(\eta))$,

$ (\eta_*)^*=((\eta_*)^*)_*=(((\eta_*)^*)_*)^*= \cdot\cdot\cdot=\omega([\eta]). $

说明2  由定理3.8 (5) 和(6) 知, 给定$I$-fuzzy拓扑生成序$\eta$.如果对$\eta$实施$_*$和$^*$运算, 则至多可以得到四种不同的$I$-fuzzy拓扑生成序, 它们分别是$\eta_*, \eta^*, \omega([\eta])$和$\omega(\iota(\eta))$.

定理3.9  设$f^{\rightarrow}:I^X\rightarrow I^Y$为满射, $(I^X, \eta)$是$I$-fuzzy拓扑生成序空间.则

(1) $(\eta/f^{\rightarrow})_*=\eta_*/f^{\rightarrow}$.

(2) $(\eta/f^{\rightarrow})^*=\eta^*/f^{\rightarrow}$.

定理3.10  设$f^{\rightarrow}:I^X\rightarrow I^Y$为双射, $(I^Y, \eta)$是$I$-fuzzy拓扑生成序空间.则

(1) $f^{\leftarrow}(\eta)_*=f^{\leftarrow}(\eta_*)$.

(2) $f^{\leftarrow}(\eta)^*=f^{\leftarrow}(\eta^*)$.

定理3.11  设$f^{\rightarrow}:(I^X, \eta)\rightarrow(I^Y, \delta)$. $(I^Y, \delta)$是弱诱导的.则$f^{\rightarrow}:(I^X, \eta)\rightarrow(I^Y, \delta)$(弱)连续当且仅当$f^{\rightarrow}:(I^X, \eta_*)\rightarrow(I^Y, \delta_*)$(弱)连续.

定理3.12  若$f^{\rightarrow}:(I^X, \eta)\rightarrow(I^Y, \delta)$ (弱)连续, 则$f^{\rightarrow}:(I^X, \eta^*)\rightarrow(I^Y, \delta^*)$(弱)连续.并且, 若$\delta$是弱诱导的, $\eta$是诱导的, 则逆定理也成立.

证  只证连续的情况. $\forall A, B\in I^Y$, 有

$ \begin{array}{l} \delta^*(A, B) = \omega(\iota(\delta))(A, B) \\ \;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}\iota(\delta)(\sigma_r(A).\sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\; = \wedge_{r\in I}\vee_{s\in I}\iota_s(\delta)(\sigma_r(A).\sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\; = \wedge_{r\in I}\vee_{s\in I}\vee\{\delta(G, H):\sigma_s(G, H)=\sigma_r(A, B)\} \\ \;\;\;\;\;\;\;\;\;\; \leq \wedge_{r\in I}\vee_{s\in I}\vee\{\eta(f^{\leftarrow}(G), f^{\leftarrow}(H)):\sigma_s(G, H)=\sigma_r(A, B)\} \\ \;\;\;\;\;\;\;\;\;\; = \wedge_{r\in I}\vee_{s\in I}\vee\{\eta(P, Q):\sigma_s(P, Q)=\sigma_r(f^{\leftarrow}(A), f^{\leftarrow}(B))\}\\ \;\;\;\;\;\;\;\;\;\; = \omega(\iota(\eta))(f^{\leftarrow}(A), f^{\leftarrow}(B))\\ \;\;\;\;\;\;\;\;\;\; =\eta^*(f^{\leftarrow}(A), f^{\leftarrow}(B)). \end{array} $

反之, $\delta(A, B)\leq\omega([\delta])(A, B) =\omega(\iota(\delta))(A, B)\leq\omega(\iota(\eta)) (f^{\leftarrow}(A), f^{\leftarrow}(B))= \eta(f^{\leftarrow}(A), f^{\leftarrow}(B))$.

定义3.2  称$I$-fuzzy拓扑生成序空间$(I^X, \eta)$是满层的, 若任意$\underline{\lambda}\in I^X$, $\eta(\underline{\lambda}, \underline{\lambda})=1$.

定理3.13   $I$-fuzzy双完全拓扑生成序空间$(I^X, \eta)$是满层的当且仅当$\omega([\eta])\leq\eta$.

证 必要性   $\forall A, B\in I^X$,

$ \begin{array}{l} \eta(A, B) = \eta(\vee_{r\in I}(\underline{r}\wedge1_{\sigma_r(A)}), \wedge_{r\in I}(\underline{r}\vee1_{\sigma_r(B)}))\\ \;\;\;\;\;\;\;\;\;\; \geq \wedge_{r\in I}\eta(\underline{r}\wedge1_{\sigma_r(A)}, \underline{r}\vee1_{\sigma_r(A)})\\ \;\;\;\;\;\;\;\;\;\;\geq \wedge_{r\in I}(\eta(\underline{r}, \underline{r}\vee1_{\sigma_r(B)})\wedge \eta(1_{\sigma_r(A)}, \underline{r}\vee1_{\sigma_r(B)})) \\ \;\;\;\;\;\;\;\;\;\; \geq\wedge_{r\in I}(\eta(\underline{r}, \underline{r})\wedge \eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)})) \\ \;\;\;\;\;\;\;\;\;\; \geq \wedge_{r\in I} \eta(1_{\sigma_r(A)}, 1_{\sigma_r(B)}) \\ \;\;\;\;\;\;\;\;\;\; = \wedge_{r\in I}[\eta](\sigma_r(A), \sigma_r(B)) \\ \;\;\;\;\;\;\;\;\;\; = \omega([\eta])(A, B). \end{array} $

充分性   $\forall\underline{\lambda}\in I^X$, $\omega([\eta])(\underline{\lambda}, \underline{\lambda})= \wedge_{r\leq\lambda}[\eta](X, X)\wedge\wedge_{r>\lambda} [\eta](\emptyset, \emptyset)=1$.因此$\eta(\underline{\lambda}, \underline{\lambda})\geq \omega([\eta])(\underline{\lambda}, \underline{\lambda})=1$.从而$(I^X, \eta)$是满层的.

定理3.14   $I$-fuzzy双完全拓扑生成序空间$(I^X, \eta)$是诱导的当且仅当它满层且是弱诱导的.

推论3.3   $I$-fuzzy双完全拓扑生成序空间$(I^X, \eta)$是可生成的当且仅当$\omega([\eta])=\eta$.

问题  如果$\eta$不是双完全的, 那么定理3.13的必要性任然成立吗?

定理3.15  设$f^{\rightarrow}:I^X\rightarrow I^Y$为双射, $(I^X, \eta)$是$I$-fuzzy拓扑生成序空间.则$(I^X, \eta)$满层当且仅当$(I^Y, \eta/f^{\rightarrow})$满层.

定理3.16   设$f^{\rightarrow}:I^X\rightarrow I^Y$为双射, $(I^Y, \eta)$是$I$-fuzzy拓扑生成序空间.则$(I^Y, \eta)$满层当且仅当$(I^X, f^{\leftarrow}(\eta))$满层.

本节中$I$-fuzzy拓扑生成序都为双完全的.文献[8-9]中定义了fuzzifying和$I$-fuzzy拓扑定义如下:

称映射$\varrho:X\rightarrow I$为$X$上的一个fuzzifying拓扑, 若$\varrho$满足

(FT1) $\varrho(\emptyset)=\varrho(X)=1$.

(FT2) $\forall U, V\subset X$, $\varrho(U\cup V)\geq\varrho(U)\wedge\varrho(V)$.

(FT3) $\forall j\in J, U_j\subset X$, $\varrho(\wedge_{j\in J}U_j)\geq\wedge_{j\in J}\varrho(U_j)$.

这时称$(X, \varrho)$一个fuzzifying拓扑空间^[8].

称映射$\zeta:I^X\rightarrow I$为$I^X$上的一个$I$-fuzzy拓扑, 若$\zeta$满足

(IFT1) $\zeta(\underline{0})=\zeta(\underline{1})=1$.

(IFT2) $\forall A, B\in I^X$, $\zeta(A\vee B)\geq\zeta(A)\wedge\zeta(B)$.

(IFT3) $\forall A_j\in I^X, j\in J$, $\zeta(\wedge_{j\in J}A_j)\geq\wedge_{j\in J}\zeta(A_j)$.

这时称$(X, \zeta)$一个$I$-fuzzy拓扑空间.有关诱导的$I$-fuzzy拓扑等相关概念参见文献[9].

定理4.1  设$\tau$是$X$上的fuzzy双完全拓扑生成序.定义$T(\tau):X\rightarrow I$为:任意$U\subset X$, $T(\tau)(U)=\tau(U, U)$.则$T(\tau)$为$X$上的fuzzifying拓扑.并且$\omega(T(\tau))=T(\omega(\tau))$.另外, 若$\tau_1\leq\tau_2$都是$X$上的fuzzy双完全拓扑生成序, 则$T(\tau_1)\leq T(\tau_2)$.

证   (FT1) $T(\tau)(\emptyset)=\tau(\emptyset, \emptyset)=1$, $T(\tau)(X)=\tau(X, X)=1$.

(FT2) $\forall U, V\subset X$,

$ \begin{array}{l}T(\tau)(U\cup V)=\tau(U\cup V, U\cup V)\geq\tau(U, U\cup V)\wedge\tau(V, U\cup V)\\ \geq\tau(U, U)\wedge\tau(V, V) = T(\tau)(U)\wedge T(\tau)(V).\end{array} $

(FT3) $\forall U_j\subset X, j\in J$,

$ \begin{array}{l}T(\tau)(\wedge_{j\in J}U_j)= \tau(\wedge_{j\in J}U_j, \wedge_{j\in J}U_j) \geq\wedge_{j\in J}\tau(\wedge_{j\in J}U_j, U_j)\\ \geq\wedge_{j\in J}\tau(U_j, U_j)=\wedge_{j\in J}T(\tau)(U_j).\end{array} $

因此$T(\tau)$是fuzzifying拓扑.另外, 任意$A\in I^X$, 有

$ \omega(T(\tau))(A)=\wedge_{r\in I}T(\tau)(\sigma_r(A)) =\wedge_{r\in I}\tau(\sigma_r(A), \sigma_r(A)) =\omega(\tau)(A, A)=T(\omega(\tau))(A). $

故$\omega(T(\tau))=T(\omega(\tau))$.其余的证明显然.

定理4.2  设$\eta$是$I^X$上的I-fuzzy双完全拓扑生成序.定义$T(\eta):I^X\rightarrow I$为:任意$A\subset X$, $T(\eta)(A)=\tau(A, A)$.则$T(\eta)$为$I^X$上的$I$-fuzzy拓扑.并且

(1) $T([\eta])=[T(\eta)]$.

(2) $T(\iota(\eta))\geq\iota(T(\eta))$.

若$\eta_1\leq\eta_2$都是$I^X$上的I-fuzzy双完全拓扑生成序, 则$T(\eta_1)\leq T(\eta_2)$.

证  只证(1) 和(2), 其余类似于定理4.1可证.任意$U\subset X$,

(1) $T([\eta])(U)=[\eta](U, U)=\eta(1_U, 1_U)=T(\eta)(1_U)=[T(\eta)](U)$.故$T([\eta])=[T(\eta)]$.

(2)

$ \begin{array}{l} T(\iota(\eta))(U)= \iota(\eta)(U, U)=\vee_{r\in I}\iota_r(\eta)(U, U) \\ \;\;\;\;\;\;\;\;\;\;= \vee_{r\in I}\vee\{\eta(A, B):\sigma_r(A, B)=(U, U)\} \\ \;\;\;\;\;\;\;\;\;\;\geq \vee_{r\in I}\vee\{\eta(A, A):\sigma_r(A)=U\}\\ \;\;\;\;\;\;\;\;\;\; =\vee_{r\in I}\vee\{T(\eta)(A):\sigma_r(A)=U\} \\ \;\;\;\;\;\;\;\;\;\;= \iota(T(\eta))(U).\end{array} $

定理4.3  若$(I^X, \eta)$是弱诱导的, 则$I$-fuzzy拓扑空间$(I^X, T(\eta))$是弱诱导的.

定理4.4  若$(I^X, \eta)$满层, 则$I$-fuzzy拓扑空间$(I^X, T(\eta))$满层.

推论4.1   若$(I^X, \eta)$是诱导的, 则$I$-fuzzy拓扑空间$(I^X, T(\eta))$也是诱导的.

定理4.5  设$f:X\rightarrow Y$为满射, $(X, \tau)$双完全, 则$T(\tau/f)=T(\tau)/f$, 且

$ \omega(T(\tau/f))=\omega(T(\tau))/f^{\rightarrow}. $

证  由定理2.11和定理4.1知, $T(\tau/f)$为$Y$上的fuzzifying拓扑.任意$U\subset Y$,

$ T(\tau/f)(U)=\tau/f(U, U)=\tau(f^{-1}(U), f^{-1}(U))=T(\tau)(f^{-1}(U))=T(\tau)/f(U). $

从而$T(\tau/f)=T(\tau)/f$.

最后, 对于任意$A\in I^Y$,

$ \begin{array}{l}\omega(T(\tau/f))(A)=\wedge_{r\in I}T(\tau/f)(\sigma_r(A))= \wedge_{r\in I}T(\tau)/f(\sigma_r(A))\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}T(\tau)\circ f^{-1}(\sigma_r(A))\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}T(\tau)(\sigma_r(f^{\leftarrow}(A)))= \omega(T(\tau))/f^{\leftarrow}(A).\end{array} $

得证.

定理4.6  设$f:X\rightarrow Y$为双射, $(Y, \tau)$双完全, 则$T(f^{-1}(\tau))=f^{-1}(T(\tau))$, 且$T(f^{\leftarrow}(\omega(\tau)))=\omega(f^{-1}(T(\tau)))$.

证  由定理2.12和定理4.1知, $T(f^{-1}(\tau))$是$X$上的fuzzifying拓扑.另外, 任意$U\subset X$,

$ T(f^{-1}(\tau))(U)=f^{-1}(\tau)(U, U)=\tau(f(U), f(U))=T(\tau)(f(U))=f^{-1}(T(\tau))(U). $

最后, 对于任意$A\in I^X$,

$ \begin{array}{l}T(f^{\leftarrow}(\omega(\tau)))(A)= f^{\leftarrow}(\omega(\tau))(A, A) =\omega(\tau)(f^{\rightarrow}(A), f^{\rightarrow}(A))\\ \;\;\;\;\;\;\;\;\;\; =\wedge_{r\in I}T(\tau)(\sigma_r(f^{\rightarrow}(A))) \\ \;\;\;\;\;\;\;\;\;\;= \wedge_{r\in I}T(\tau)(f(\sigma_r(A))) \\ \;\;\;\;\;\;\;\;\;\;=\wedge_{r\in I}f^{-1}(T(\tau))(\sigma_r(A)) =\omega(f^{-1}(T(\tau)))(A).\end{array} $

定理4.7  设$f^{\rightarrow}:I^X\rightarrow I^Y$满射, $(I^X, \eta)$是$I$-fuzzy双完全拓扑生成序空间.则$T(\eta/f^{\rightarrow})=T(\eta)/f^{\rightarrow}$是$I^Y$上$I$-fuzzy拓扑.

定理4.8  设$f^{\rightarrow}:I^X\rightarrow I^Y$双射, $(I^Y, \eta)$为$I$-fuzzy双完全拓扑生成序空间.则$T(f^{\leftarrow}(\eta))=f^{\leftarrow}(T(\eta))$是$I^X$上的$I$-fuzzy拓扑.

定理4.9  若$f^{\rightarrow}:\hskip-1mm(I^X, \eta_1)\rightarrow(I^Y, \eta_2)$(弱)连续, 则$f^{\rightarrow}:\hskip-1mm(I^X, T(\eta_1))\rightarrow(I^Y, T(\eta_2))$(弱)连续.